Understanding adjoint functors

Solution 1:

I will tell you how I remember if something is a left or right adjoint. Hopefully it's useful for you.

Let $\mathcal{C},\mathcal{D}$ be categories, and let $F:\mathcal C \to \mathcal D$, $G:\mathcal D \to \mathcal{C}$ be functors.

By definition $F$ is left-adjoint to $G$ if there are natural isomorphisms $$\overline{(\ )}:\mathcal{D}(FA, -) \to \mathcal{C}(A,G-)$$ $$ \overline{(\ )}:\mathcal{C}^{\mathrm op}(GB,-) \to \mathcal{D}^{\mathrm op}(B,F-) $$

for all objects $A \in \text{ob}\mathcal C$ and $B \in \text{ob}\mathcal D$, such that they are mutual inverses when you plug $B$ in the top one and $A$ in the bottom.

The way to remember that $F$ is a left adjoint is that in the first nice covariant natural transformation, $F$ is on the left.

So your diagram is simply the naturality square for the first transformation: hence $F$ is the left adjoint in that case.

EDIT OVER A YEAR LATER: An easier way to say the above is $F$ is left-adjoint to $G$ if there is a natural isomorphism $$ \mathcal D( F-_1, -_2) \cong \mathcal C(-_1, G-_2) $$ of functors $\mathcal C^{\text{op}} \times \mathcal D \longrightarrow \mathsf{Set}$.

Solution 2:

Given functors $F:\mathcal{D} \to \mathcal{C}$ and $G:\mathcal{D} \to \mathcal{C}$ with natural bijections $\text{hom}_\mathcal{C}(F(X),Y) \to \text{hom}_\mathcal{D}(X,G(Y))$ we say that $F$ is left adjoint to $G$. Thus tensor product is left adjoint to the Hom functor. I guess this naturally makes sense because in the defining equations the functor $F$ is on the left and $G$ on the right.

In terms of the proof, the map you have written down is correct, but of course one should actually show that everything works; i.e. that your map $\eta_{ab}$ is a bijective and that is is natural (i.e. that the diagram commutes).

It is also fairly standard to write $\text{Hom}_R(f \otimes \text{id}_M,g)$ as $(f \otimes \text{id}_M)^*$

Edit: Please see Bruno's comment below. For a map $f:A \to A'$ and a fixed $B$ it is normal to write $(f \otimes \text{id}_M)^*$. Otherwise $\text{Hom}_R(f \otimes \text{id}_M,g)$ seems to be the correct thing to write. (Note that in your question you need to change the $B$ in the lower left hand corner of the commutative diagram to a $B$')